The invention relates to a process and an apparatus for the reconstruction of three-dimensional images of an object involving measurements on two circular paths having a common axis, i.e. the paths belong to parallel planes and their centres are joined by an axis perpendicular to the plane of the paths. If the planes of the paths coincide, they are concentric. It is also possible to envisage more numerous paths.
The reconstruction of images relates to a given object to be examined and the means used comprise two dimensional arrays of sensors each traversing one of the paths or, in an equivalent manner, a single array which traverses all the paths.
The image of the object is defined by values assumed by a function on each of its points. The function is also a property of a radiation (e.g. X or gamma) having a conical shape and with a focal point and which passes through the object. Each ray is received by one of the sensors of the bidimensional array and consequently represents the sum of the function on all the points of the object belonging to said ray. An appropriate processing of the sums on all the rays for an adequate number of measurements in accordance with different incidences around the object makes it possible to reconstitute the image of the object.
In practice, consideration is only given to finite numbers of rays and points of the object in accordance with discretizations or interconnections.
The present invention relates to an improvement to an earlier invention described in European patent application EP-A-0292 402, but it is possible to envisage the use of tile present invention in other circumstances or using other mathematical data processing methods.
The methods which can be envisaged in particular use what is called the Radon transform of the function to be measured, which is defined as the sum of the function on each of the planes, called Radon planes, which pass through the object in question or, even better, the primary derivative of said transform. The contribution of the points of these planes which do not belong to the object are considered to be non-existent, which is valid in the case of a radiation passing through an ambient gaseous medium without being attenuated. Here again, a discretization is performed in order to only perform the calculations on a finite number of planes.
The primary derivative of the Radon transform is defined as being the derivative of the Radon transform as a function of the variable .rho. defining the distance from the plane in question to an origin. For each plane, it corresponds to the sum on said plane of the primary derivative of the Radon transform in the direction perpendicular to the plane. The invention described in EP-A-0292 402 demonstrates that for a plane passing through one position of the focal point of the radiation cone and encountering the bidimensional array of sensors, it is possible to calculate on the basis of measurements the exact value of the primary derivative of the Radon transform on said plane. This calculation, in accordance with the formulas given in EP-A-0292 402, makes use in preferred manner of a focal point distance correction weighting, two filtering operations corresponding to the calculations of the primary derivatives respectively along the rows and columns of the array of sensors, two summations along the intersection line between the plane to be processed and the bidimensional array of sensors and then a linear combination and standardization of the results. This summation makes use of the necessary interpolations, because these intersection lines pass between the rows of sensors or intersect them.
Throughout the remainder of the application, the sum is understood to mean the weighted sum of the measured values (in the case of the Radon transform), as well as the linear combination of the sums of weighted and filtered values (in the case of the primary deriratire of the Radon transform) obtained along the array rows and columns.
The sum of the function on the points of the Radon planes is easy to obtain, provided that these planes have an intersection with the bidimensional array of sensors and pass through the single focal point aimed at by the sensors. It is sufficient to form the sum of the values measured by each sensor located at the intersection, with the necessary interpolations, because the intersections of the Radon planes pass between rows of sensors or intersect them. Once the values of the function on the Radon planes have been calculated, inversion formulas exist and which are described in the aforementioned patent application, which make it possible to arrive at the values of the function on the points of the interconnection of the object corresponding to the image to be reconstructed.
However, it is necessary to return to conditions making it possible to obtain an adequate number of Radon planes to permit a satisfactory description of the object. Each Radon plane can be defined by what is called its characteristic point, i.e. the projection point on said plane of an arbitrarily chosen origin 0. This characteristic point, designated C in FIG. 1, can be defined by its spherical coordinates .rho., .phi. and .theta. of ray, longitude and colatitude respectively on the basis of the origin 0. The Radon plane P passing through the characteristic point C can be defined by the radius .rho. and the unitary vector n of direction OC.
The values of the function on the Radon plane can only effectively be calculated for the Radon planes intersecting the path covered by the focal point of the radiation. In the case of an attenuation function, said focal point is specifically a point source of X, gamma and similar rays. This is the concept which is described in the aforementioned European patent application. The same geometrical conditions exist in emission tomography, when the function to be measured is the activity emitted by the body. The focal point then has no physical existence and simply corresponds to the convergence point towards which all the collimators used in front of the bidimensional array are directed. In order to obtain an adequate number of Radon planes, it is necessary to carry out several measurements with different positions of the focal plane.
On considering a circular path T covered by the focal plane (or the source) S, the planar array of sensors Pdet passing through the same path as the focal point S or possibly a concentric path with a different radius and it will also be assumed that the origin 0 used for defining the characteristic point C is located on the axis of the path T. The volume enveloping the characteristic points corresponds to a torus To produced by the rotation of a spherical surface of diameter OS around the rotation axis of the path T. Thus, the Radon planes passing through the focal point S have their characteristic points distributed over the spherical surface of diameter OS, because the angle SCO is a right angle. The torus To is called the characteristic volume of the measurements, which thus depends on the shape of the path T and its position relative to the origin 0.
It is also sufficient, in order to obtain a complete description of the object by processes using the sums of the function on the Radon planes, to be able to have characteristic points belonging to a characteristic volume of the object, which is always included in a sphere V centered on the origin 0 and which envelops the object M. Thus, it is certain that if the characteristic volume of the object is included in the characteristic volume of the measurements, it will be possible to reconstruct the image of the object M. In the case of a circular path T, this condition is unfortunately not fulfilled, because a shadow area remains for characteristic points of the sphere V not belonging to the torus To and whose Radon planes do not intersect the path T.